Capacitated vehicle routing problem with pickup and alternative delivery (CVRPPAD): model and implementation using hybrid approach
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Abstract
The paper presents an optimization model and its implementation using a hybrid approach for the Capacitated Vehicle Routing Problem with Pickup and Alternative Delivery (CVRPPAD). The development of the CVRPPAD was motivated by postal items distribution issues. The model proposed combines various features of Vehicle Routing Problem variants. The novelty of this model lies in the introduction of alternativeness of item delivery points, differentiation of delivery point types in terms of their capacity (parcel lockers) and possibility of collecting the items during the execution of delivery routes. The model also takes into account a possibility of introducing time windows related to delivery time. The proposed data structure in the form of sets of facts facilitates the implementation of the model in all environments, Constraint Logic Programming (CLP), Mathematical Programming (MP), metaheuristics, databases, etc. The model is implemented in the MP, hybrid CLP/MP, and hybrid CLP/heuristic environments. The hybrid CLP/MP approach is the authors’ original solution, which has already been used to solve Supply Chain Management problems, scheduling problems, routing problems etc. For large size problems, considering their combinatorial character, the proposed CLP/MP approach is ineffective. Its effectiveness will improve when MP is replaced by a heuristic. All implementations were performed on the same data instances (facts), which made it possible to compare them according to the solution search time, number of decision variables and number of constraints. The directions into which the model can be further developed were also presented.
Keywords
VRP CVRP Constraint logic programming mathematical programming Hybrid approach Optimization Heuristics1 Introduction
Observed in the last 10 years, fast development of information technologies, online services and mobile applications has resulted in a significant growth of ecommerce segment. The base of the European ecommerce customers who have ordered goods or services over the Internet for private use (2014) ranges from 15% in Bulgaria, 30% in Poland and Latvia and more than 75% in the UK and Denmark (Eurostat http://ec.europa.eu/eurostat/statisticsexplained/index.php/Main_Page). Despite rapid expansion, there is still room for further development and the ecommerce is rising in impact (Eurostat http://ec.europa.eu/eurostat/statisticsexplained/index.php/Main_Page). This impact has already had its effect in the area of postal items distribution. Ecommerce, enjoying no limits of time and space, increased the number and volume of international postal items. Postal service providers are required to provide high quality, efficient and reliable services.
Another element, outside the global ecommerce, that affects the postal service market is the emergence of a number of postal and courier service operators and a declining market share of local operators. This process started more than 20 years ago in Western Europe and about 10 years ago in the east European countries.
Rising competition among postal operators has contributed to the development of new types of services, such as parcel lockers, mail tracking systems with realtime tracking information sent to a customer’s mobile, etc. Universal service providers have undergone major organizational changes. Currently, RFID tags for itemlevel tracking are being looked at as an alternative technology to barcodes. The main advantage of this technology is that it is easy to automate data capture process at shipment and sorting hubs.
That said, these changes have failed to solve or even deepened existing problems distributors have to face with respect to the socalled first and last mile (Zellner et al. 2016) in their supply chains. These problems are related to finding the optimal solution in terms of cost and time for both delivery and collection of postal items to/from end customers. And today’s definition of a customer may include an individual (home delivery), parcel lockers or a postal outlet.
The main motivation of this research was to build a model for the optimization of postal items distribution (letter and parcel distribution with the use of universal couriers), which would take into account certain currently existing conditions and limitations. The conditions are different from those in the previous models (VRPs, CVRPs, etc.). These new conditions include: a new type of customer with limited capacity (parcel lockers), an opportunity to take alternative delivery points, one point of delivery supported by several couriers, delivery and pick up at the same time and etc. For solving this model, an innovative, original hybrid approach is proposed, which gives a far greater modeling flexibility and efficiency [already tested in modeling and optimization of SCM (Sitek et al. 2014), 2ECVRP (Sitek 2014) etc.].
The rest of this paper is structured as follows. Section 2 presents the literature review concerning VRPs. Description and formalization of the presented problem is described in Sect. 3. Section 4 explains the proposed implementation approach based on integrated CLP/MP and CLP/heuristic environments. The computational results for numerical experiments are presented in Sect. 5. A summary of conclusions and future works is presented in Sect. 6.
2 Vehicle routing problems: literature review

to minimize all transportation cost associated with the used means of transport (e.g. vehicles) as well as the distance travelled;

to minimize the number of means of transport (e.g. vehicles) needed to cover all delivery points;

if the estimated travel time and vehicle load have been exceeded, to minimize this overrun.

The items are delivered from a depot (one or more).

A set of vehicles is operated by a set of drivers/couriers.

Given is the road/train network.

Given is a set of delivery points where the items are delivered.

VRPPD (Vehicle Routing Problem with Pickup and Delivery): In this variant, a number of items need to be moved from certain pickup points to other delivery points. The VRPPD objective is to find a set of optimal routes for the means of transport to visit the dropoff and pickup points.

VRPTW (Vehicle Routing Problem with Time Windows): The delivery points have time windows within which the deliveries must be made.

CVRP (Capacitated Vehicle Routing Problem): The means of transport have limited carrying capacity. Another CVRP variant, CVRPTW, has time windows.

2ECVRP (TwoEchelon Capacitated Vehicle Routing Problem): Depotdelivery points deliveries pass through intermediate points (called satellites).

VRPMT (Vehicle Routing Problem with Multiple Trips): The means of transport can do more than one route.

OVRP (Open Vehicle Routing Problem): The means of transport do not have to return to the depot.

VRPB (Vehicle Routing Problem with Backhauls) : There are two types of delivery points, deliveries (linehauls) and pickups (backhauls). There are additional constraints: (a) each means of transport must perform all the deliveries before making any pickups; (b) routes with only backhauls are disallowed.
The most often used methods and approaches for solving VPRs can be divided into exact approaches (mathematical programming, integer programming, etc.), heuristics, constructive methods, twophase algorithms and metaheuristics (ant algorithms, constraint programming, genetic algorithms, tabu search, simulated annealing etc.) (Yeun et al. 2008; Archetti and Speranza 2014).
The exact methods used for VRPs include mainly the BranchandBound algorithm and its variants e.g., BranchandPrice, BranchandCost, and BranchCutandPrice. The BranchCutandPrice exact algorithm was used in Ropke et al. (2007) for a CVRP with twodimensional loading constraints. In GutiérrezJarpa et al. (2010), the exact BranchandPrice algorithm was proposed for solving a multiple vehicle routing problem with time window—VRPTW. Numerous exact column generation or DantzigWolfe decomposition algorithms were proposed that can accommodate complex constraints and timedependent costs (Azi et al. 2010; Rousseau et al. 2004). None of those methods has proved capable of dealing with solving VPRs of appreciable size at acceptable time, hence the need to use approximate methods in the form of heuristics and metaheuristics. Among multiple heuristics dedicated to individual VRPs, the most important include the clusterfirst, routesecond (Fisher and Jaikumar 1981), neighborhood search heuristics to optimize the planned routes of vehicles in the context where new requests, with a pickup and a delivery location occur in real time (Gandreau et al. 2006), a route construction heuristics with an adaptive parallel scheme (Peng 2011), Granular Tabu Search (GTS) (Toth and Vigo 2014), etc. Nearly all of these methods are heuristic and metaheuristic approaches due to the NPhardness of the VRP. The Constraint Logic Programming (CLP) (Apt and Wallace 2006; Rossi et al. 2006) approach to many problem types involves building a solution incrementally, backtracking when infeasibility is detected, until a solution is found or the problem is proven to have no solution. This approach can be applied to a variety of VRPs, but it is usually too inefficient to deal with realworld cases. In most realworld cases, the existence of a solution is never in doubt. The quality of the solution is essential and, therefore, CLP is usually used to solve VRP subproblems or evaluate the feasibility of routes and time slots, available capacity etc. (Rossi et al. 2006).
In some situations, when an exact (optimal) solution is needed, exact methods are used, supported by some heuristics or metaheuristics. Metaheuristics can be applied to finding initial solutions or as presolving methods for narrowing solution spaces or eliminating infeasible solutions. In such cases we can talk about hybrid methods. Motivation behind this work was to construct a reference model for the problem of postal item distribution to final delivery points through universal couriers.
In addition to building the reference model, this contribution includes the implementation and solution of the model with the use of the hybrid approach that applies CLP for initial solving and MP or heuristics for finding the final optimal solution. The universal information structure, based on the sets of facts, was also proposed for the reference model. The structure can be applied to the entire class of VRPs and implemented with ease as databases, XML flat files, etc.
3 Capacitated vehicle routing problem with pickup and alternative delivery (CVRPPAD): problem description
The objective is to minimize the distances for couriers to travel and also the “penalty” for delivering items to alternative points. This is the most obvious objective function chosen for practical reasons, but other options are possible, for example, the number of trips/couriers or the shortest delivery time, etc.

Three categories of delivery recipients/points are considered: i.e., postal outlet (post office, parcel service point, etc.), home delivery (individual customer), parcel locker (PackStation, PACCOMAT, MyQuickBox, etc.)

The fleet of vehicles is heterogeneous (volume/capacity).

A vehicle may perform no more than one trip in a planning period.

Vehicles’ capacity constraints are imposed.

Parcel lockers’ capacity constraints are imposed.

Alternative delivery locations are possible.

Each route between delivery points has a specified travel time and the overall travel time length cannot be greater than the preset travel time length.

Items cannot reloaded.

Each courier’s route starts from and ends at the depot.

Note—No constraints are imposed on route length, number of delivery points on the route, customers’ priorities, however the model is flexible enough to add these constraints if needed.
3.1 Problem formulation
Indices, parameters and decision variables
Symbol  Description 

Indices  
P  A set of delivery points /e.g. parcel locker, postal outlet, etc./ 
I  A set of all items 
C  A set of all couriers 
p,j  Delivery point index (p, j \(\in \) P) 
i  Item index (i\(\in \)I) 
c  Courier index (c\(\in \)C) 
d  Depot /hub/ 
Parameters  
\(\hbox {vi}_{\mathrm{i}}\)  Item volume (volumetric weight) i (i\(\in \)I) 
\(\hbox {vc}_{\mathrm{c}}\)  Courier’s vehicle volume c (c\(\in \)C) 
\(\hbox {co}_{\mathrm{p}}\)  Number of items that point p can accept (p\(\in \)P) 
\(\hbox {de}_{\mathrm{ip}}\)  If item i can be delivered to delivery point p, then de\(_{ip}\,=\,1\), otherwise de\(_{ip}\) = 0 (i\(\in \)I, p \(\in \) P) 
\(\hbox {di}_{\mathrm{pj}}\)  Distance between delivery points p, j, (p, j\(\in \)P\(\cup \)d) 
\(\hbox {pe}_{\mathrm{ip}}\)  Penalty for delivery of item i to point p(i\(\in \)I, p\(\in \)P) 
\(\hbox {ti}_{\mathrm{pj}}\)  Transfer time between points p, j, (p, j\(\in \)P\(\cup \)d) 
\(\hbox {dr}_{\mathrm{i}}\)  If item i is delivered from the depot to the delivery point then dr\(_{i}\) = 1 otherwise dr\(_{i}\) = 0 (i\(\in \)I) 
T  Non extendible delivery time frames 
A  Arbitrarily large constant 
Decision variables  
\(\hbox {X}_{\mathrm{cpj}}\)  If courier c travels from point p to point j, then \(X_{cpj}\,\)= 1, otherwise \(X_{cpj}=0,\)(c\(\in \)C, p, j\(\in \)P\(\cup \)d) 
\(\hbox {Y}_{\mathrm{cpji}}\)  If courier c travels from point p to point j carrying item i, then \(Y_{cpji}\,\)= 1, otherwise \(Y_{cpji}\) = 0, (c\(\in \)C, p,j\(\in \)P\(\cup \)d) 
\(\hbox {E}_{\mathrm{ic}}\)  If item i is delivered by courier c, then \(E_{ic}\,=\,1\), otherwise \(E_{ic}\) = 0 (i\(\in \)I, c\(\in \)C) 
\(\hbox {Z}_{\mathrm{ip}}\)  If item i is delivered to the pointp, then \(Z_{ip} =1,\) otherwise \(Z_{ip}\) = 0 (i\(\in \)I, p\(\in \)P) 
Meanings of the constraints
Constraints  Description 

(2)  Arrival and departure of a courier at/from delivery point (parcel locker, postal outlet, and individual customer) 
(3)  If no items are to be carried on the route, a courier does not travel that route 
(4)  If a courier does not travel along a route, no items are to be carried on that route 
(5)  At no route segment courier carries more items than allowable vehicle volume 
(6)  Items are delivered to one of possible delivery points 
(7)  One run of a courier 
(8)  Item delivered to only one delivery point 
(9)  Non extendible delivery point volume/At the point—the number of items matches the number of locations/ 
(10)  Items picked up/delivered to a delivery point 
(11)  Runs executed within the required time 
(12)  Each courier picks up/ delivers items from/to a source (depot) 
(13)  Assignment of a courier to an item 
(14)  A given item can be assigned to one courier 
(15)  Binarity 
4 Implementation
Due to the character of the proposed model (BLP), mathematical programming (MP) provides a natural environment for its implementation. A number of solvers are available, both commercial (LINGO, CLPEX and free of charge (SCIP, LP_Solve), which use MP methods such as simplex, branch&bound, branch&price, etc. (Schrijver 1998). These methods are exact methods, which, taking into account a binary character of decision variables and characteristics of VRPs (NPhard), renders them inefficient especially in the case of real data instances of larger sizes. Thus based on their previous experience in solving similar problems (Sitek et al. 2014; Sitek 2014; Sitek and Wikarek 2015b), the authors propose a hybrid approach to the implementation and solving (optimization) of the model (1)–(15).
Here, the hybrid approach involves combining two environments, MP and CLP, in the following way. The CLP environment with builtin mechanisms for solving constraints (e.g., constraint propagation, backtracking, etc.) Apt and Wallace (2006) and Rossi et al. (2006) is used in the presolving phase. In this phase, the values of some of the decision variables can be established, the domains of other variables can be narrowed and some constraints can be aggregated and transformed before the actual solving process starts. The presolving phase diagram for the model (1)–(15) is shown in Sect. 4.2 (Fig. 3). The model is solved in the second phase with the use of an MP method. Another element of the implementation is the design of a suitable information structure for the model (Sect. 4.1). The universal information structure allows integrating the model implementation with equivalent commercial environments Enterprise Resource Planning (ERP), Distribution Resources Planning DRP), Warehouse Management System (WMS), Transportation Management System (TMS) and implementing the model with the use of data bases or XML files. The information structure is presented in the form of sets of facts, as in Fig. 2. The hybrid approach with CLP used for presolving is a universal idea and can be combined with other environments outside MP, such as Genetic Algorithm (GA), Ant Colony Optimization ACO, etc.
4.1 Information structurethe sets of facts
4.2 Presolving
Presolving is used to reduce the dimensions of the model and, consequently, provides a possibility of solving other largesize models within acceptable time. Figure 3 shows the diagram outlining the presolving stages for the (1)–(15) model. In the first stage, the facts F_item, F_delivery are transformed. All postal items can be grouped according to a category, e.g., a delivery point and an item type, which reduces the number of postal items to handle (groups instead of single postal items are considered). In the next stage, parameter (de\(_{ip})\) is used to establish which items (groups) have a delivery point alternative. After the analysis of this variable, some of decision variables \(Z_{ip}\) take the value of 0 or 1, whereas the remaining variables that can take the value 1 and 0 are not established at this stage (their values are established during the optimization stage). On the basis of constraints (6) and (12), the values of those decision variables \(Y_{cpji}\) for which \(Z_{ip }\)have been established in the earlier stage are determined. The final presolving stage involves determining the values of \(X_{cpj}\) on the basis of constraints (3) and (4) and decision variables \(Y_{cpji }\)of preestablished values.
4.3 Heuristic for solving CVRPPAD
For the problems of larger sizes, considering their combinatorial character, the proposed hybrid approach which integrates MP and CLP is insufficiently effective. Therefore, a heuristic is proposed (Fig. 4), which can replace MP in the hybrid approach. The heuristic is dedicated to the structure and properties of problems such as CVRPPAD. It does not guarantee finding an optimal solution, but in combination with CLP, it provides an effective and efficient tool. The heuristic is characterized by three selection criteria/priorities for R1, R2, R3 parameters (Table 3). Being very flexible, this approach can be used according to the character of data. The travelling salesman problem was solved with the known from the literature variants of NN (Nearest Neighborhood) methods (Gandreau et al. 2006; Peng 2011; Regoa et al. 2011).
In short, the main idea of the heuristic algorithm (Fig. 4) is that in the first step, a courier is chosen based on the R1 criterion and the deliveries and their recipients are determined based on criteria R2 and R3. In the next step, the route for the courier is set (in accordance with the NN method).
The heuristic can be used for solving CVRPPAD in two ways, either directly or as an element of the hybrid approach, in which it will replace the MP solver. Its unquestionable advantage consists of the reduced computation time and the possibility of solving the models of larger sizes within acceptable time.
Possible values of criterion
Criterion  Possible values 

R1  Courier with the highest capacity 
Courier the smallest capacity  
R2  Delivery point which has the largest capacity 
Delivery point which has the smallest capacity  
Delivery point with the largest number of items  
Delivery point with the smallest number of items  
Delivery point with the largest capacity of items  
Delivery point with the smallest capacity of items  
R3  First, the items that do not have an alternative delivery points 
First, the items with bigger capacity from the delivery point 
Results for numerical experiments P1...P12
P  Lpr  Lpu  Lku  MPbased approach  Hybrid1 approach (CLP and MP)  

V  C  T  \({\hbox {F}_\mathrm{C}}\)  V  C  T  \(\hbox {F}_\mathrm{C}\)  
(a)  
MPsolverLINGO  
P1  20  5  2  1672  404  234  128  580  294  4  128 
P2  20  5  2  1672  404  454  39  580  294  7  39 
P3  20  5  2  1672  404  346  139  580  294  10  139 
P4  20  8  4  5243  872  87  230  2946  2124  45  230 
P5  40  8  4  10443  1283  345  258  2946  2124  67  258 
P6  60  8  4  15543  1523  600\(^{**}\)  258\(^{*}\)  2946  2124  36  258 
P7  80  8  4  20642  1813  600\(^{**}\)  NFSF  2946  2124  39  262 
P8  100  8  4  25743  2093  600\(^{**}\)  NFSF  2946  2124  42  262 
P9  1000  100  20  35674567  20056  2400\(^{**}\)  NFSF  267804  456747  987  56455 
P10  1000  200  20  56675643  45567  2400\(^{**}\)  NFSF  456456  774564  1545  97865 
P11  2000  100  20  75434534  56576  2400\(^{**}\)  NFSF  267804  456747  878  56455 
P12  2000  200  20  143456467  87565  2400\(^{**}\)  NFSF  456456  774564  2134  104343 
MPsolverSCIP  
P1  20  5  2  1672  404  185  128  580  294  6  128 
P2  20  5  2  1672  404  302  39  580  294  8  39 
P3  20  5  2  1672  404  245  139  580  294  9  139 
P4  20  8  4  5243  872  81  230  2946  2124  41  230 
P5  40  8  4  10443  1283  301  258  2946  2124  54  258 
P6  60  8  4  15543  1523  574  258  2946  2124  31  258 
P7  80  8  4  20642  1813  600\(^{**}\)  456\(^{*}\)  2946  2124  39  262 
P8  100  8  4  25743  2093  600\(^{**}\)  NFSF  2946  2124  41  262 
P9  1000  100  20  35674567  20056  2400\(^{**}\)  NFSF  267804  456747  876  56455 
P10  1000  200  20  56675643  45567  2400\(^{**}\)  NFSF  456456  774564  1321  97865 
P11  2000  100  20  75434534  56576  2400\(^{**}\)  NFSF  267804  456747  794  56455 
P12  2000  200  20  143456467  87565  2400\(^{**}\)  NFSF  456456  774564  1934  104343 
P  Lpr  Lpu  Lku  Heuristics  Hybrid2 approach (CLP and Heuristics)  

T  \(\hbox {F}_\mathrm{C}\)  T  \(\hbox {F}_\mathrm{C}\)  
(b)  
P1  20  5  2  23  128  12  128  
P2  20  5  2  23  42\(^{*}\)  12  39  
P3  20  5  2  23  154\(^{*}\)  12  139  
P4  20  8  4  23  260\(^{*}\)  12  230  
P5  40  8  4  26  324\(^{*}\)  13  260\(^{*}\)  
P6  60  8  4  28  258  13  258  
P7  80  8  4  28  294\(^{*}\)  15  274\(^{*}\)  
P8  100  8  4  29  276\(^{*}\)  14  286\(^{*}\)  
P9  1000  100  20  47  67434\(^{*}\)  22  56455  
P10  1000  200  20  49  124332\(^{*}\)  22  97967\(^{*}\)  
P11  2000  100  20  53  67531\(^{*}\)  22  56674\(^{*}\)  
P12  2000  200  20  52  264563\(^{*}\)  24  104343 
5 Numerical experiments
In order to verify the proposed model as well as evaluate the hybrid approach, we made a number of computational experiments (P1...P12) with instances of the facts contained in Appendix A (due to the limited volume of the paper, complete data instances were included only for P1...P8). The experiments varied in the number of couriers (up to 20 couriers), delivery points (5–200), items (up to 2000), types of transport (up to 3 modes of transport), and in the type of delivery points (2).
To implement the models and carry out computational experiments, Eclipse (www.eclipse.org) was used as part of the CLP, while Lingo (www.lindo.com) and (http://scip.zib.de/) were used as MP solvers.
In the first stage of the experiment, examples P1...P12 were implemented and solved in four approaches, MP, hybrid1 (CLP/MP with presolving) heuristics and hybrid2 (CLP/Heuristics with presolving), with no time windows included. The results are shown in Table 4a, b and the detailed results for P1, P2, P3 and P8 in Table 5 (Appendix B). The optimal solution (hybridbased approach) for P8 is shown in Fig. 5 and the feasible solution (MPbased approach) in Fig. 6 and in Table 6 (Appendix B). From these data, it follows that the use of the hybrid1/hybrid2 approach and heuristics made it possible to find a solution in a much shorter time.
Analysis of the results clearly shows the superiority of hybrid approach. The hybrid approach (hybrid1) as an exact method reduces calculation time by the order of 2 to 65 times compared to the MP method. However, using different solvers, MP is not so significant. SCIP is approximately 20% more effective than LINGO.
The hybrid approach (hybrid2) provides double reduction in the computation time and improves the quality of approximate solutions (0–10% worse than optimal) in relation to heuristics (the quality of approximate solutions are 0–150% worse than optimal).
6 Conclusions
This paper introduced a new VRP variant called CVRPPAD (Capacitated Vehicle Routing Problem with Pickup and Alternative Delivery), a crucial tool for postal items distribution, city logistic, etc. The CVRPPAD has unique features that distinguish it from other variants of the VRPs (Sect. 3). The most important characteristics include taking into account the capacity of delivery points and alternative locations. These changes are due to the inclusion of a new type of delivery point—a parcel locker. The proposed model of the CVRPPAD combines the features of many different variants of VRP such as delivery and pickups, vehicle capacity, different type of vehicles and so on.
The model was tested with the classical MP environment, “LINGO” and “SCIP”, with the proposed hybrid approach combining two environments, MP and CLP (Bockmayr and Kasper 2004; Milano and Wallace 2010; Sitek and Wikarek 2016; Hooker 2002; Sitek and Wikarek 2015a; Silva 2001), and with hybrid approach combining CLP and proposed heuristics (Sect. 4.3). Declarative CLP was used to presolve the model. MP and heuristics were used for the final model solving (optimization). The use of this integrated approach has reduced the size of the model by the factor of almost 10, thus reducing significantly the optimization time and, for the larger examples, finding the solution within acceptable time span, as opposed to the MP implementations. The proposed hybrid approach, the presolving method in particular, can be used for the integration with heuristics and metaheuristics necessary to solve the CVRPPAD model of industrial size.
The proposed model and method of its solution can be widely implemented in information management systems of postal items distribution. For this purpose, a universal information structure is proposed in the form of sets of facts that are easily integrated with the databases. Further research will follow two directions.
The model will be expanded to cover a route length, number of points on the route, items prioritization, lead times etc. (Nielsen et al. 2014b, a) and implementation in the cloud (Deniziak et al. 2013). The model is planned to be enriched with fuzzy logic (Bocewicz et al. 2016).
The authors declare that there is no conflict of interests regarding the publication of this paper.
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